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A vital part of Jacob Lurie's classification of fully extended topological field theories [1], very roughly, says that any representation of the n-Cobordism category $Z: {\rm Cob}_{{n}} \to C$ depends …

Undergraduate-Level Background Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A …

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice vers …

1. Algebra from action groupoids Let $G$ be a finite group acting on a finite set $X$ from the right (denoted in element as $x^{g}$). We have an algebra (of the action groupoid) over $\mathbb{C}$: the …

$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the Jack "$J$" polynomials [1]. The latter have profound relations with representation theory. For example, …

Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this doe …

TL; DR. In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I won …

For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in t …

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\rank{rank}$Let $G$ be a finite group, and $C=\Rep(G)$ be the monoidal category of complex finite-dimensional representations of $G$. As $C$ is finite …

I think it's quite natural. $W$ is built up by simple reflections $s_i$. KL involution is the $\mathbb{Z}$-linear map sending $\delta_{s}$ to $\delta^{-1}_{s^{-1}}$ and $v$ to $v^{-1}$.. you basicall …

This question is motivated by Why do combinatorial abstractions of geometric objects behave so well? The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials Kazhdan-Lusztig-Stanley polynomial …

Question Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-unde …

This is a study note that spells out @Konstantinos's answer explicitly. Preface Our goal is to classify all finite dimensional representations over the complex number field for the quantum double …

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However …

I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure. Let $G$ be a finite group. Its finite-di …